# Properties

 Label 53371a Number of curves $3$ Conductor $53371$ CM no Rank $1$ Graph # Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("a1")

sage: E.isogeny_class()

## Elliptic curves in class 53371a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
53371.b3 53371a1 $$[0, -1, 1, 1873, -2003]$$ $$32768/19$$ $$-421122861451$$ $$[]$$ $$48048$$ $$0.91997$$ $$\Gamma_0(N)$$-optimal
53371.b2 53371a2 $$[0, -1, 1, -26217, -1729538]$$ $$-89915392/6859$$ $$-152025352983811$$ $$[]$$ $$144144$$ $$1.4693$$
53371.b1 53371a3 $$[0, -1, 1, -2161057, -1222057453]$$ $$-50357871050752/19$$ $$-421122861451$$ $$[]$$ $$432432$$ $$2.0186$$

## Rank

sage: E.rank()

The elliptic curves in class 53371a have rank $$1$$.

## Complex multiplication

The elliptic curves in class 53371a do not have complex multiplication.

## Modular form 53371.2.a.a

sage: E.q_eigenform(10)

$$q + 2q^{3} - 2q^{4} - 3q^{5} - q^{7} + q^{9} + 3q^{11} - 4q^{12} - 4q^{13} - 6q^{15} + 4q^{16} - 3q^{17} - q^{19} + O(q^{20})$$ ## Isogeny matrix

sage: E.isogeny_class().matrix()

The $$i,j$$ entry is the smallest degree of a cyclic isogeny between the $$i$$-th and $$j$$-th curve in the isogeny class, in the Cremona numbering.

$$\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)$$

## Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)

The vertices are labelled with Cremona labels. 